GCF Calculator - Greatest Common Factor of Numbers

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest positive integer that divides each of a given set of integers without a remainder. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 exactly. The two most common algorithms for computing the GCF are the Euclidean algorithm and prime factorization. The Euclidean algorithm is the more efficient of the two for large numbers. It works by repeatedly replacing the pair (a, b) with (b, a mod b) until the remainder is 0; the last nonzero value of b is the GCF. The algorithm runs in O(log min(a,b)) steps, making it extremely fast even for very large integers. Prime factorization computes the GCF by expressing each number as a product of primes raised to powers, then taking the product of each prime raised to the minimum power found across all numbers. For example, 12 = 2^2 * 3 and 18 = 2 * 3^2, so GCF(12, 18) = 2^1 * 3^1 = 6. While less efficient than the Euclidean algorithm for large numbers, prime factorization provides clear pedagogical insight into why the GCF is what it is. The GCF has many practical applications. In arithmetic, it is used to reduce fractions to their simplest form: to simplify a/b, divide both numerator and denominator by GCF(a, b). In geometry, the GCF of two lengths gives the longest ruler that measures both without a remainder. In computer science, the GCF appears in modular arithmetic, cryptographic algorithms (such as RSA key generation), and data compression. For more than two numbers, the GCF is computed iteratively. GCF(a, b, c) = GCF(GCF(a, b), c). This calculator handles any number of positive integers and supports both the Euclidean algorithm (for fast results) and prime factorization (for detailed step-by-step output). The prime factorization view is particularly useful for students learning about factors and divisibility.